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Graph Theory
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Undergraduate
By
rizwanhudda
on Aug. 24, 2012 | Updated Dec. 6, 2017
Second Minimum spanning tree
Given an weighted undirected graph \( G = (V, E)\), and \(w : E \mapsto R^+\). Let T be MST i,e minimum spanning tree of graph G. Second MST is a Tree T' different from T, and its weight is less t…
Computer Science
Mathematics
Algorithms
Graph Theory
trees
0
Undergraduate
By
John Doe
on Nov. 3, 2012 | Updated Dec. 6, 2017
Direct reduction from Graph Homomorphism to SAT
The decision problem \(GraphHomo\) is defined as follows: \(GraphHomo = \{\langle G, H\rangle \mid \text{there is a graph homomorphism from G to H}\}\) Give a direct reduction from \(GraphHomo\) to …
Computer Science
Mathematics
Complexity Theory
Graph Theory
Logic
np
reduction
sat
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Undergraduate
By
Chandra Chekuri
on July 29, 2012 | Updated Dec. 6, 2017
Simple path containing three given nodes
Let \(G=(V,E)\) be an undirected graph. Describe a linear time algorithm that given \(G\) and three distinct nodes \(u,v,w\) decides whether there is a simple path in \(G\) that contains all of them.
Computer Science
Mathematics
Algorithms
Graph Theory
linear time algorithms
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Undergraduate
By
diego
on June 27, 2012 | Updated Dec. 6, 2017
Unmatchable edges of bipartite graphs
Prove that the following algorithm finds the unmatchable edges of a bipartite graph \(G\) (edges that aren't in any perfect matching): find a perfect matching in \(G\), orient the unmatched edges from…
Mathematics
Graph Theory
bipartite graph
matching
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Undergraduate
By
Chandra Chekuri
on July 29, 2012 | Updated Dec. 6, 2017
Diameter and low-degree vertex
Let \(G = (V,E)\) be an undirected connected graph. Suppose \(G\) has a pair of nodes \(s,t\) that are distance \(d\) apart. Show that there is a vertex \(v\in G\) such that the degree of \(v\) is at…
Computer Science
Mathematics
Algorithms
Graph Theory
counting
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Undergraduate
By
diego
on June 8, 2012 | Updated Dec. 6, 2017
The cube of a connected graph is hamiltonian
Prove that the vertices of any connected graph \(G\) can be listed in a cyclic order so that the distance in \(G\) of every two consecutive vertices is at most \(3\). Moreover, show that this can be …
Computer Science
Mathematics
Algorithms
Graph Theory
hamiltonian cycle
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Graduate
By
Shiva Kintali
on June 17, 2012 | Updated Dec. 6, 2017
Vertex cover using DFS
Consider the following algorithm for Vertex Cover of a graph \(G\) : Run depth first search (DFS) on \(G\). Output the vertices which are not leaves in the DFS tree. Prove the following : The ou…
Computer Science
Mathematics
Approximation Algorithms
Graph Theory
vertex cover
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Undergraduate
By
Shiva Kintali
on June 7, 2012 | Updated Dec. 6, 2017
Caterpillars are Graceful
Graceful Labeling : Label the vertices of a simple undirected graph \(G(V,E)\) (where \(|V| = n\) and \(|E| = m\)) with integers from \(0\) to \(m\). Now label each edge with the absolute difference o…
Mathematics
Graph Theory
graph labeling
0
Graduate
By
Shiva Kintali
on Aug. 8, 2012 | Updated Dec. 6, 2017
Gallai Identities
Consider the following parameters of an undirected graph \(G\) on \(n\) vertices. \(\nu(G)\) is the size of a maximum matching of \(G\). \(\tau(G)\) is the size of a minimum vertex cover of \(G\). …
Mathematics
Graph Theory
edge cover
independent set
matching
vertex cover
0
Undergraduate
By
Shiva Kintali
on May 31, 2012 | Updated Dec. 6, 2017
Party Problem
Suppose there are six people at a party. Prove that there are always three of them so that every two know each other (or) no two know each other. In other words, let the edges of the complete graph o…
Mathematics
Puzzles
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counting
extremal graph theory
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